The prayer you refer to is taken from the Eucharistic liturgy and is said by the priest as he pours water and wine into the chalice. It doesn’t mean the Church teaches we are or shall become God himself.
Sometimes Catholic theologians refer to the "divinization of man," but this is nothing like New Age teaching. When Scripture says that Christians are "partakers of the divine nature (2 Pt 1:4) it means we begin to share in God’s divine life of grace in this world. We become divinized when God dwells within us.
This sharing in the divine nature, though real, is a created participation in God’s life. As creatures we can never become the uncreated God.